On Wikipedia we are trying to update the data on Mercury's distance from Earth since Mercury has a varying eccentric orbit. Last year I used Horizons to generate "Mercury Closest Approaches to Earth from 1950 to 2200" showing that currently Mercury does not come closer than 0.5489AU (82 million km): http://home.comcast.net/~kpheider/Mercury.txt (see bottom section)

Under http://en.wikipedia.org/wiki/Mercury_(planet)#Observation (cut and paste link) someone using Solex 10 suggests, "In 871, the nearest approach was the first in about 41,000 years to be closer than 82.2 Gm, something that has happened 68 times since then, as of 2008. After much longer gaps, the next approach to within 82.1 Gm is in 2679, and to 82 Gm in 4487. But it will not be closer to Earth than 80 Gm until 28,622." (Edited)

Can GravSim generate a plot of the distance of Mercury from Earth over say a 50,000 year period? This would hopefully help illustrate some concepts. -- Kevin

Difficult question to plot a graph for this time period autmatically ! I guess JPL ends at 2200. I gave it a try using the Taylor integrator ( Picard) . After some manipilation with Excel I got the following graph for the next 1000 years . The output was split in timeframes of 10 years starting at 1 feb 2007 . The minimum value of each 10 year period ( if smaller then before ) is represented in the graph . So the next close approach in the coming 10 years will be 31th may 2015 . It's nice to see how the close approaches seem to gather each time beginnig of june . The minimum value I got at the end is 82.080.535.160 m

Edit : I see the figure herunder gives a match in the year 2679 with the Solex Integration

This is a really tricky task Kevin . It requires a very good accurancy to perform this . Not only the orbits should be integrated very accurate in terms of orbital elements , but also the true anomaly should be very precise for both bodies . If this isn't the case the close approaches will miss. I've been to quick to run the above graph , as I noticed that runnig it at a higher accurancy the results vary a little . The graph herunder was run at an accurancy 10 times higher than the one above . Contrary to the Solex integration which was mentionned above the minimum distance keeps decreasing after 2679 . I think its hard to be confident about results after more than a couple of hundred years ( lets say thousand year ) given the accumulating errors .

Ahh. This is why I like multiple sources. The Solex 10 comment on the Mercury article had a few typos. The comment has been changed to say, "After much longer gaps, the next approach to within 82.1 Gm is in 2679, and to 82 Gm in 4487. But it will not be closer to Earth than 80 Gm until 28,622."

Hereunder a graph of Mercury's distance to Earth which I ran at high accurancy . This result is I think in accordance with the Solex 10 simulation mentionned above . The distance seems to be decreasing with time . Reasons for this behaviour are variations in the eccentricity of Mercuries orbit , the precession of its orbit , but also the change in inclination ....But also Earths orbit changes in this time period . I'll try to post some additional info about this variations .

The decreasing distance between Mercury and Earth is partially due to the changing inclination and eccentricity of Mercuries orbit ( see graph below) . But also Earths orbit changes . In this timeperiod the inclination of Earth rises to more than 2°.

As a final picture of Earth's relation to Mercury an animation which clearly shows the dynamics of both orbits : The orbits seem to be static at first sight , till the animation starts again . Clearly visible is the precession of the perihelion of Mercury , but also Earth orbit seems to shift upwards. The animation was run over 22000 years . Unfortunately due to downsizing the sun isn't visible anymore in this screenshots .

The decreasing distance between Mercury and Earth is partially due to the changing inclination and eccentricity of Mercuries orbit ( see graph below) . But also Earths orbit changes . In this timeperiod the inclination of Earth rises to more than 2°.

Wow Frank. Those are very nice charts. I am impressed. I can really see how Mercury gets closer since the inclination drops as the eccentricity increases. -- Kevin

Kevin , as asked hereby the evolution of Earths Inc and Ecc in comparison with Mercurys over a timespan of more than 26000 years . Graph was generated using the Picard Integration . The result should be comparable with the results you get with the Solex10 program . The broad variation in Ecc of Earths orbit is due to our Moon . The result is also availabe in .xls format .

It took some time to have the full solar system simulated for a period of 250.000 years from "now" . But here's the plot of the innermost planet ... Inclination reaches a maximum of 10° , while eccentricity is generally decreasing . Something happens around 100.000 years from now for the Sma. Edit : above simulation was run under Newtonian gravity and does not take into account any oblateness .

Some part of the mystery of Venus' inclination may become clear if we take a look at the following picture: Earth and Venus are more closely related than I've previously thought . Their inclinations vary in counterphase . The period seems to be around 70.000y